Operational Resource Theory of Coherence

TL;DR

This paper develops an operational framework for quantum coherence, defining coherence distillation and cost, and providing simple formulas for these quantities in the asymptotic limit, revealing irreversibility in coherence processing.

Contribution

It introduces an operational resource theory of quantum coherence with explicit formulas for distillation and cost, and characterizes reversible states within this framework.

Findings

Distillable coherence equals the relative entropy of coherence.

Coherence cost is given by the coherence of formation.

The theory is generally irreversible, with no bound coherent states.

Abstract

We establish an operational theory of coherence (or of superposition) in quantum systems, by focusing on the optimal rate of performance of certain tasks. Namely, we introduce the two basic concepts - "coherence distillation" and "coherence cost" in the processing quantum states under so-called incoherent operations [Baumgratz/Cramer/Plenio, Phys. Rev. Lett. 113:140401 (2014)]. We then show that in the asymptotic limit of many copies of a state, both are given by simple single-letter formulas: the distillable coherence is given by the relative entropy of coherence (in other words, we give the relative entropy of coherence its operational interpretation), and the coherence cost by the coherence of formation, which is an optimization over convex decompositions of the state. An immediate corollary is that there exists no bound coherent state in the sense that one would need to consume coherence to create the state but no coherence could be distilled from it. Further we demonstrate that the coherence theory is generically an irreversible theory by a simple criterion that completely characterizes all reversible states.